Conjecture on the nef divisor and Waldschmidt constant of the Klein configuration

Let $X_\mathcal K$ be the blowup associated with the Klein line configuration, with $H$ the pullback of a line and $E_4,E_3$ the exceptional divisors over its quadruple and triple points. Let $I_\mathcal K$ be the homogeneous ideal of the singular points of the configuration. Klein Waldschmidt-constant conjecture. The divisor

$$D=28H-2E_4-5E_3$$

is nef, and consequently

$$\widehat\alpha(I_\mathcal K)=\frac{13}{2}.$$

The claim is motivated by computational evidence and bounds obtained from classifying negative curves in bounded degree; its general validity remains open.