Let $\,q\,$ be a prime of the form $\,4\, m_q+3$.

I ask if it is always possible to find two primes $\,p_1$ and $\,p_2$ of the form $\,4\, m_p+1$ such that $$q=\sqrt{p_1+\sqrt{p_2+q}}$$ E.g. $$3=\sqrt{5+\sqrt{13+3}}$$ $$7=\sqrt{37+\sqrt{137+7}}$$ $$11=\sqrt{101+\sqrt{389+11}}$$ $$19=\sqrt{337+\sqrt{557+19}}$$ $$23=\sqrt{521+\sqrt{41+23}}$$