  We consider the Cauchy problem for an energy supercritical nonlinear wave
equation that arises in $(1+5)$--dimensional Yang--Mills theory. A certain
self--similar solution $W_0$ of this model is conjectured to act as an
attractor for generic large data evolutions. Assuming mode stability of $W_0$,
we prove a weak version of this conjecture, namely that the self--similar
solution $W_0$ is (nonlinearly) stable. Phrased differently, we prove that mode
stability of $W_0$ implies its nonlinear stability. The fact that this
statement is not vacuous follows from careful numerical work by Bizo\'n and
Chmaj that verifies the mode stability of $W_0$ beyond reasonable doubt.
