  If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove
that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2\lambda_0\in(0,1)$ a.s. on
$\{X_t\neq 0\}$, where $-\lambda_0\in(-1,-1/2)$ is the lead eigenvalue of a
killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller,
Mytnik and Perkins who proved the above with positive probability. To establish
this result we derive some new basic properties of a recently introduced
boundary local time and analyze the behaviour of $X(t,\cdot)$ near the upper
edge of its support. Numerical estimates of $\lambda_0$ suggest that the above
Hausdorff dimension is approximately $.224$.
