Say we have a non-random matrix $\textbf{K}\in\mathbb{R}^{p\times p}$ with $\text{rank}(\textbf{K})=\rho\leq p$ and a random projection $\textbf{R}\in\mathbb{R}^{d\times p}$ with i.i.d. standard Gaussian entries. The projection dimension $d$ is allowed to vary. The question is: for what range of $d$ is the quantity $$\textbf{R}\textbf{K}\textbf{R}^T$$ non-singular?
This was how I thought about it: for the $d\times d$ matrix $\textbf{R}\textbf{K}\textbf{R}^T$ to be invertible, I need it to have a rank of $d$. By restricting $d<p$, I ensure that $\text{rank}(\textbf{R})=d$ almost-surely. Using the inequality $$\text{rank}(\textbf{R}\textbf{K}\textbf{R}^T)\leq \text{min}(\text{rank}(\textbf{R}),\text{rank}(\textbf{K}\textbf{R}^T)),$$ along with this fact, I have $\text{rank}(\textbf{R}\textbf{K}\textbf{R}^T)\leq d$. But I want this to hold with equality.
I know that $$C(\textbf{R}\textbf{K}\textbf{R}^T)\subseteq C(\textbf{R}\textbf{K}) \subseteq C(\textbf{R}).$$ I was thinking that to have equality I would need to show that for some range of $d$, $$C(\textbf{R}\textbf{K}\textbf{R}^T)=C(\textbf{R}\textbf{K})=C(\textbf{R}).$$ I am stuck and don't know if this is even the correct approach.
Edit: My approach is wrong as $\text{rank}(\textbf{K}\textbf{R}^T)$ in the first step does not equal $d$ in general.