Let $G$ be a game with finitely many players and $\underline{v}= (\underline{v}_i)$ be the minmax payoff profile.
Denote by $G_{\infty}(\delta)$ the infinitely repeated game whose stage game is $G$ and discount factor $\delta$. (The payoffs of $G_{\infty}(\delta)$ is ranked by the average discounted criterion.)
Claim If there exists a $\delta^*$ such that $G_{\infty}(\delta^*)$ has a SPNE with outcome $\underline{v}$, then, for all $\delta$, $G_{\infty}(\delta)$ has a SPNE with outcome $\underline{v}$.
Is the claim true or false?
I would argue that:
The claim is true if it is weakened to "...for all $\delta$, $G_{\infty}(\delta)$ has a NE with outcome $\underline{v}$." This is because, on the equilibrium path, the hypothesis implies that every player is being minmaxed.
The claim is true if we have the following additional assumption: the outcome of every subgame is $\underline{v}$, not just on the equilibrium path. This would imply that every player is being minmaxed at every history in the given SPNE of $G_{\infty}(\delta^*)$, which makes it a SPNE for any $G_{\infty}(\delta)$.
Without this strengthened assumption, I am not sure. I have no counterexample but, conceivably, there could be a subgame where a player has incentive to deviate if $\delta \ne \delta^*$.
Here is an argument by induction that the claim is true.
First, the hypothesis implies that every subgame located on the path must have the minmax outcome $\underline{v}$. Subgame perfection mean every player $i$ must get at least $\underline{v}_i$. If a player $i$ get strictly more than $\underline{v}_i$, that means he gets strictly less than his minmax payoff in the stages before and would have incentive to deviate from a previous stage, contradicting subgame perfection.
Now consider a subgame located at a history that is a one-player one-stage deviation off the path. By the above paragraph, the payoff of every player must be equal to his minmax payoff, otherwise that player would deviate from the equilibrium path to this subgame.
Therefore, by induction, any subgame that is a finite-players finite-stages deviation (i.e. any subgame) must yield minmax outcome. This implies that every player is being minmaxed at every history, which makes the strategy profile a SPNE for any discount factor.