Polynomial upper-bound conjecture for the smallest singular value of submatrices

Let $F=[g_1,g_2,\dots,g_{n+k}]$ be an $n\times(n+k)$ matrix, and define

$$\tau=\min\{\sigma_n(F_S):S\subset\{1,\dots,n+k\},\ |S|=n\}.$$

Assume that $\|g_j\|\leq1$ for every $j$. Polynomial upper-bound conjecture. There exists a constant $C=C(k)$ such that

$$\tau\leq\frac{C}{n^{k-\frac12}}.$$

For $k=1$, the source notes a previously established upper bound of order $n^{-3/2}$; the conjecture proposes the corresponding order $n^{-(k-1/2)}$ for general fixed $k$.