The enhanced Target Collision Resistance (eTCR) game implies that an adversary is allowed to choose a target message $M$. Upon choosing this target message, the attacker learns the function $\mathcal{H}_{K}$ (by learning the key $K$) and wins after presenting a new message $M'$ and a (possibly new) key $K'$ such that $H_{K}(M) = H_{K'}(M')$.

Is there a formal notion/definition similar to this, but instead the adversary wins if able to present a new key $K'$ such that $H_{K}(M) = H_{K'}(M)$?

What I'm asking here is slightly different than the (weaker) TCR notion where the attacker, under this same setup, wins after presenting a new message $M'$ such that $H_{K}(M) = H_{K}(M')$.


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