optimization - Sharpe Maximization under Quadratic Constraints

When doing Sharpe optimization

$$ \max_x \frac{\mu^T x}{\sqrt{x^T Q x}} $$

there is a common trick (section 5.2) used to put the problem in convex form. You add a variable $\kappa$ such that $x = y/\kappa$ choose $\kappa$ s.t. $\mu^T y=1$. Changing the problem to the simple convex problem

$$ \min_{y,\kappa} y^T Q y \; \text{where} \; \mu^T y = 1, \kappa > 0 $$

which is easy to solve.

Unfortunately, my problem also has a second-order constraint that becomes non-convex in $(y,\kappa)$ $$ x^T P x \leq \sigma^2 \implies y^T P y \leq \kappa^2 \sigma^2 $$

Is there a trick to keep this problem convex and allow the use of second-order cone programming algorithms?