Suppose that $X_1,X_2,\ldots$ are independent random variables. Denote $F_t$ the $\sigma$-algebra generated by $X_1,\ldots,X_t$. Let $T, M \geq 1$, and let $f_1,\ldots,f_M$ be $\mathbb{R} \to \mathbb{R}$ functions such that for every $m =1,\ldots, M$, $t =1,\ldots, T$, $E[f_m(X_t) | F_{t-1}]= 0$. Denote for every $t, m$, $S_{m,t} = \sum_{s=1}^t f_m(X_t)$.

Suppose in addition that there exists $\sigma^2 < \infty$ such that for every $t, m$, $E[f_m^2(X_t) | F_{t-1}] \leq \sigma^2$.

For any fixed $m$, we know from Kolmogorov's inequality for martingales that, for any $x > 0$ $$P\left[ \max_{t=1,\ldots, T} \frac{S_{m,t}}{\sqrt{T}} > x \right] \leq x^{-2} \sigma^2.$$

Do we have something like $$P\left[\max_{m=1,\ldots,M} \max_{t=1,\ldots, T} \frac{S_{m,t}}{\sqrt{T}} > x \right] \leq C x^{-2} \sigma^2$$ for a constant that doesn't depend on $M$?