Strong subgame perfect equilibria

A $k$-player (turn-based) game is given by a directed graph $G = (V, E)$ with each vertex owned by one of the $k$ players, and an objective $W_i \subseteq V^\omega$ for each player $i$. A strategy profile $\bar{\sigma} = (\sigma_1, \ldots, \sigma_k)$ is a tuple of strategies, one for each player. A strategy profile is Nash equilibrium (NE) if no player can benefit by unilaterally changing their strategy, i.e., if $\bar{\sigma}$ produces a losing outcome for player $i$, then Player $i$ cannot deviate to a different strategy $\sigma_i'$ that would produce a winning outcome. A strategy profile is a subgame perfect equilibrium (SPE) if it is an NE in every possible subgame of the original game. It is well-known that SPE is a more appropriate notion in this setting and well-studied for $k$-player games with $\omega$-regular objectives. A more robust notion of NE is strong Nash equilibrium (SNE), where no coalition of players can cooperatively deviate in a way that strictly benefits all of its members. Based on SNE, we consider the notion of strong subgame perfect equilibrium (SSPE), which is a strategy profile that is SNE in every subgame. Question: Does it always exists an SSPE in $k$-player games with $\omega$-regular objectives? If yes, can we compute a maximal SSPE?