  Suppose $\Lambda$ is a special biserial algebra over an algebraically closed
field. Schr\"oer showed that if $\Lambda$ is domestic then the radical of the
category of finitely generated (left) $\Lambda$-modules is nilpotent, and the
least ordinal, denoted $\mathrm{st}(\Lambda)$, where the decreasing sequence of
powers of the radical stabilizes satisfies $\mathrm{st}(\Lambda)<\omega^2$.
With Gupta and Sardar, the third author conjectured that if $\Lambda$ has at
least one band then $\omega\le\mathrm{st}(\Lambda)<\omega^2$ even when
$\Lambda$ is non-domestic. In this paper we settle this conjecture in the
affirmative. We also describe an algorithm to compute $\mathrm{st}(\Lambda)$ up
to a finite error. We also show that for each $\omega\leq\alpha<\omega^2$ there
is a finite-dimensional tame representation type algebra $\Gamma$ with
$\mathrm{st}(\Gamma)=\alpha$.
