Limiting-function conjecture for normalized $k$-mer minimizer counts

Let $\Sigma$ be the underlying alphabet, and for each $k,m$ let $f_{k,m}:[0,1]\to[0,1]$ be the piecewise linear function determined by the points $p(w)=(x,y)$ for $w\in\Sigma^m$, with $f_{k,m}(x)=y$, together with $f_{k,m}(0)=1$ and $f_{k,m}(1)=0$. Limiting-function conjecture. There exists a function $f_\infty:[0,1]\to[0,1]$ such that $(f_{k,m})$ converges to $f_\infty$ in a sense to be determined as $k,m\to\infty$. The conjecture records numerical evidence of a common limiting profile, but the mode of convergence and the existence of the limit remain unspecified.