  We study the total positivity of the multiplicative convolution kernel T
associated with the independent product of two random variables $B(a,b)$ and
$\Gamma(c).$ This kernel is totally positive of infinite order if $b$ or $d =
a+b -c$ are integers. Otherwise the sign-regularity of T has always a finite
order, which is here computed. More precisely, for every $n\ge 1$ it is shown
that T is totally positive of order $n + 1$ if and only if $(d,b)$ lies above a
certain stairway ${\mathcal E}_n$ plotted in the upper half-plane. This
stairway also characterizes the sign-invariance of several determinants
associated with the confluent hypergeometric function of the second kind.
