Sokal's reversed second-order Eulerian polynomial conjecture

For $r\geq1$, define the $r$th-order Eulerian polynomials by $E_0^{[r]}(x)=1$ and, for $n\geq1$,

$$E_n^{[r]}(x)=xA_n^{(r)}(x)=\sum_{k=0}^{n-1}\genfrac{\langle}{\rangle}{0pt}{}{n}{k}^{(r)}x^{k+1}.$$

Define the reversed polynomial by

$$E_n^{[r]*}(x)=x^nE_n^{[r]}(1/x).$$

Sokal's reversed second-order Eulerian conjecture. The sequence $(E_n^{[2]*}(x))_{n\geq0}$ is coefficientwise Hankel-totally positive in $x$.

The ordinary $r$th-order Eulerian polynomials already have coefficientwise Hankel-total positivity via a branched Stieltjes-type continued fraction. The assertion for the reversed second-order sequence is presented as a separate conjecture, with no resolution given in the source.