Steuding's universality conjecture for Selberg-class functions

Let $S$ be the Selberg class, let $\mathcal L\in \widetilde{\mathcal S}\cap S$, let $K$ be a compact subset of the strip $\mathfrak S=\{\sigma_m<\sigma<1\}$ with connected complement, and let $g$ be a continuous function on $K$, holomorphic in $K^0$, with no isolated zeros in $K^0$. Steuding's conjecture. For every $\epsilon>0$,

$$\liminf_{T\to\infty}\frac{1}{T}\operatorname{meas}\left\{t\in[0,T]:\max_{s\in K}|\mathcal L(s+it)-g(s)|<\epsilon\right\}>0.$$

This is a stronger measure-theoretic universality statement than the preceding theorem and is intended to provide information about the Riemann Hypothesis. For the Riemann zeta-function, Johan Andersson showed that it is equivalent to a conjecture on zero-free polynomial approximation.