I think it does but I'm having trouble showing it.
Using the integral test: the function $f(x)=\frac{1}{\ln(x^x+x^2)}$ is decreasing but I'm having trouble integrating $\int_{1}^{\infty} \frac{dx}{\ln(x^x+x^2)}$, trying the substitution $t=x^x$ didn't seem to work.
The root test gives $\sqrt[n]{\left |\frac{1}{\ln(n^n+n^2)}\right |} \to 1$ which doesn't imply anything.