The question you were asked is quite good. Some applicants will simply fail the question in some way. Like you did. Others will give a correct answer but not recognize anything beyond the obvious asked by the question. Still others will go further and say something more that will make them "stand out in a crowd." And the extra things that can be said, or asked, about this circuit are myriad. So it's really pretty good. I may have to steal it. ;)
The circuit, at \$t= 0\$, looks like:
simulate this circuit– Schematic created using CircuitLab
There is a circuit loop present there. It's the loop formed by \$C_1\$, \$C_2\$, and \$R_1\$. (There is no circuit loop involving \$V_1\$, so no current in \$V_1\$.)
By inspection, you can see that there is \$6\:\text{V}\$ across \$R_1\$ at \$t=0\$. This means that a current through \$R_1\$ must exist at \$t=0\$. But this current is everywhere the same in the circuit loop. So the same current found in \$R_1\$ will also be found going equally through \$C_1\$ and \$C_2\$.
So there will be some manner of change in the voltages across each capacitor for some time, until such time as the voltage across \$R_1\$ is effectively zero. (It's never exactly zero, but gets arbitrarily close to zero for all intents and purposes.)
The above circuit is what I'd see, immediately. And I think you also saw it, too. But you didn't recognize that \$V_x\$ would change. I don't know why you didn't think about it. But apparently you didn't consider the change over time that they wanted to see from you.
I would have taken out a piece of paper and marked a place \$+5\,\tau\$ out on the x-axis, after first computing \$\tau=R_1\cdot\frac{C_1\,C_2}{C_1+C_2}=\frac23\:\text{s}\$. Then I would have marked on the y-axis a point at \$+3\:\text{V}\$ and \$+9\:\text{V}\$(point A). I would then mark a point \$+3\:\text{V}\$ above \$+5\,\tau\$(point B). Now I would draw out an RC decay from A to B, making sure to hit about 37% (about 1/3rd) of the way above \$+3\:\text{V}\$, between that and the \$+9\:\text{V}\$ value at \$+1\,\tau\$. Also then about 1/8th of the way above at \$+2\,\tau\$. The rest would have been more freehand. Like this:
(It is hard to draw with a table mouse! My pen tablet is elsewhere right now.)
But something else would have caught my attention. The differing sizes of the two capacitors. Sure, I can combine them like I did, above, for the purposes of the question at hand. But another question would have intruded, as well. What would the final voltages on them be?
I'd quickly see that one is twice the size of the other, so that the current would have to yield a rate of change for \$C_1\$ that is twice as fast as that for \$C_2\$. (This is what the basic capacitor equation says.) So then I'd think, "Hmm. Both start with the same voltage across them, but one changes twice as fast as the other one. Since I know at the end of time there will be no current in \$R_1\$ then their sum must be zero. But what about their final values?"
Then I'd think, "Well, on average they will each lose 3 V. But that's an average. What would average to 3 V but where one change is twice the other?" And then I'd know. It must be a \$4\:\text{V}\$ change on one of them and a \$2\:\text{V}\$ change on the other one. Of course, the smaller change would be on the larger one, \$C_2\$. So then I'd know that the final voltages will be \$C_1\$ with \$+3\:\text{V}-4\:\text{V}=-1\:\text{V}\$ and \$C_2\$ with \$+3\:\text{V}-2\:\text{V}=+1\:\text{V}\$.
And then I'd be able to quip something about the fact that, at the end, one of these large capacitors would have a reversal of polarity and that this might be a concern.
It also provides me with the tools I may need should they then decide to ask about the voltages across the capacitors.
This question allows an interviewee to say more than just what the question asks. And they could also follow up and ask what you asked, which is whether or not there would be a current in the voltage source. (No.) Just to see if the interviewee shows any confusion about that, as well.
It is a good question. Simple and allowing simple answers. But complex enough to allow someone who sees further to show them they do. I like it.