The starred Dixmier's conjecture

Dixmier's famous question says the following: Is every algebra endomorphism of the first Weyl algebra, $A_1(F)$, where $F$ is a zero characteristic field, an automorphism? Let $α$ be the exchange involution on $A_1(F)$: $α(x)= y$, $α(y)= x$. An $α$-endomorphism of $A_1(F)$ is an endomorphism which preserves the involution $α$. Then one may ask the following question, which may be called the "$α$-Dixmier's problem $1$" or the "starred Dixmier's problem $1$": Is every $α$-endomorphism of $A_1(F)$ an automorphism?

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