# (key) Target Collision Resistance (kTCR)?

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 $$H_{K}(M) = H_{K}(M')$$.