Let $L$ be the language of all even-length strings whose first half is a palindrome.
Let $L$ be the language of all even length strings whose first half is imbalanced—with an unequal number of $\mathtt{a}$'s and $\mathtt{b}$'s. I believe $L$ is not context-free; the intuition is that the equipment of a PDA can either be used to keep both halves of the string the same length, or to construct complex internal structure like a palindrome or an imbalanced string, but not both simultaneously.
I would like to formalize this intuition as a general conjecture, beyond this particular example $L$. I suggest that, letting $A\bowtie B$ denote the language $\{xy : x\in A,\;y\in B,\; |x|=|y|\}$,
Conjecture: $A\bowtie \Sigma^*$ is context free if and only if $A$ is regular.
This conjecture would then establish that $L\equiv \mathsf{imbalanced}\bowtie \Sigma^*$ isn't context-free. One direction $(\Leftarrow)$ is easy, using regular grammars. But I wonder if anyone knows or can see a proof of the other direction?
Is a proof of this statement already well known or easily constructible?
Maybe it would be easier to prove this more broadly:
Conjecture: If $A\bowtie B$ is context free and $A$ and $B$ have the same word sizes $\{|x| : x\in A\} = \{|y| : y \in B\}$, then $A$ and $B$ are regular.
So far I have tried arguing about the derivation tree for $A\bowtie \Sigma^*$, considering cases of which parentheses are allowed on which halves of the string in the Chomsky Schuetzenberger theorem, limiting to the case where $A$ and $\Sigma^*$ have different(-colored) alphabets, and similar. I have proved it for the case where the alphabets of $A$ and $B$ are disjoint (e.g. color-coded), by arguing about the cases of colors of strings that the nonterminals of a grammar for $A\bowtie B$ can produce in CNF.
Edit: Having looked at generating functions, I know that if $A(x)$ is the generating function for strings of each length $n$ in $A$, then $A(2x^2)$ is the generating function for $A\bowtie \Sigma^*$, and I know that regular grammars correspond to rational generating functions, unambiguous context-free grammars have algebraic generating functions, and $\bowtie$ corresponds to the Hadamard product, which has certain closure properties relative to rational and algebraic functions—but since I don't know a characterization of what kind of context-free languages $A\bowtie B$ can be, I don't see what to uncover next.
