Payoff imbalance in Nash equilibria of the imbalanced $(m,2k+1)$-RPS game

Consider the given imbalanced $(m,2k+1)$-RPS game and a Nash equilibrium $N$. Let $P_i(S)$ be the probability that player $i$ plays $S$, and let $P_i(T_p)$ be the probability that player $i$ plays any $P$-type object. Imbalance conjecture. For every player $i$,

$$\frac{P_i(T_p)}{P_i(S)}\geq m-1.$$

Moreover, for any $k>1$ and player $q_i$, the expected number of other players tying with $q_i$ for the win occurring for $P_k$ is less than $m-2$. The conjecture is intended to support a proof that the constructed imbalanced $(m,2k+1)$-RPS game is strongly playable; the source offers only a heuristic based on the rapidly decreasing probability of playing $S$ in the symmetric three-object Nash equilibrium.