Minimalist conjecture for central vanishing of level-one cusp forms

Let $f$ be a cuspidal Hecke eigenform for the group $\mathrm{SL}_2(\mathbb Z)$ of weight $k$, and let

$$m(f)=\operatorname{ord}_{s=1/2}L(s,f).$$

Minimalist central-vanishing conjecture. If $k\equiv2\, (\mathrm{mod} {\, 4})$, then $m(f)=1$. When $k\equiv2\, (\mathrm{mod} {\, 4})$, the functional equation forces central vanishing, so the conjecture asserts that this forced zero is simple and that there is no additional vanishing. The source says that no precise conjecture for this case had previously been recorded, and cites positive-proportion results for nonzero central derivatives as evidence.