Linear upper-bound conjecture for the largest singular value

Let $\sigma_{1,n}$ denote the largest singular value of the matrix associated with index $n$. Largest-singular-value bound.

$$\sigma_{1,n} \leq \sqrt{2.0193n-0.7914}.$$

This bound is suggested by the numerical homotopy experiment and is presented as a conjecture for the growth of the largest singular value.