Conjectured stability for unequal block sizes

Let the conditions of the stability theorem hold, but now for block sizes $n_i$ that are not necessarily equal. Define the mean block size and variance by

$$\bar{n} = \frac{1}{m}\sum_i n_i, \qquad \sigma_n^2 = \frac{1}{m}\sum_i (n_i - \bar{n})^2.$$

If the relative standard deviation satisfies $\sigma_n / \bar{n} \ll 1$, then conjectured stability for unequal block sizes. the stability condition holds approximately with $n$ replaced by $\bar{n}$. Specifically, if

$$m < C(\gamma, F_X, K) \cdot \bar{n}^{1+\frac{\gamma}{2}}$$

holds with a sufficient margin, the Hessian matrix $\mathbf{Q}$ remains positive definite. This extends the equal-block-size stability result perturbatively to block sizes with small relative variability; the statement is presented as a possible result for future analysis, so its validity remains open.