  We present a proof of a conjecture of Q. Shi, Y. Wang and H. Zuo claiming
that the maximal spectral number of a hypersurface isolated singularity does
not belong to the Tjurina spectrum. This follows from the self-duality of the
Jacobian ring, which is compatible with the action of $f$ and also with the
$V$-filtration. We also provide a sufficient condition for the generalized
Hertling conjecture on the variance of Tjurina spectrum to fail, and calculate
some examples using some codes in Singular.
