Mellin midpoint-value conjecture for the dimension-8 limiting functions

Let

$$M_f(s)=\int_0^\infty f(x)x^{s-1}\,dx$$

and define $M_{\widehat f}$ analogously for the Fourier transform $\widehat f$. Normalize the limiting functions by $f(0)=\widehat f(0)=1$. Mellin midpoint-value conjecture. For $n=8$,

$$M_f(4)=M_{\widehat f}(4)=\frac1{15}.$$

The equality of the two Mellin values follows from Fourier–Mellin symmetry, but the value $1/15$ is conjectural; the analogous dimension-24 value is not identified in the source.