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.