The quotient modified diagonal conjecture

Let $X$ be a projective variety over a field ${\mathbb K}$ with a nontrivial involution $\iota\in\operatorname{Aut}(X)$, let $Y:=X/\langle\iota\rangle$, and let $f\colon X\to Y$ be the quotient map. Suppose there is a fixed point $a\in X({\mathbb K})$, and write $b:=f(a)$. For modified diagonal cycles, write $A\equiv B$ when they agree in the rational Chow group.

The quotient modified diagonal conjecture. If

$$\Gamma^{m}(Y;b)\equiv 0,$$

then

$$\Gamma^{2m-1}(X;a)\equiv 0.$$

This is a proposed induction principle for transferring vanishing of modified diagonal cycles from a quotient to a double cover. The supplied context states the result as a conjecture but gives no evidence of a general proof or disproof.