Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields

Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $\mathrm{\Psi }$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le -j<k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $\kappa$ over $K$ which factor through the ${\mathbb{Z}}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{\kappa \mathrm{\Psi }},k+j)$ is a constant independent of $\kappa$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.