Conjecture on an optimal nonuniform partition for Pascal matrix recursion

Let $n \gg k > 2$, and let $\lambda_1,\ldots,\lambda_k \in \mathbb{Q}$ satisfy $\lambda_i>0$ for each $i$ and

$$\lambda_1+\cdots+\lambda_k=1.$$

Set $n_i=\lambda_i n$ for each $i$, so that the $\lambda_i$ determine a partition of $n$. Partition-runtime conjecture. There exists a choice of $\lambda_1,\ldots,\lambda_k$ such that the resulting runtime satisfies

$$T_n=\Theta(k n\log^2 n).$$

The preceding analysis establishes this bound for the uniform partition, while the paper notes that numerical experiments suggest nonuniform partitions may perform better; whether an optimal partition can attain this asymptotic bound remains unresolved.