This is a follow up to What is the current status on the corank conjecture for Selmer groups?
Let E be an elliptic curve over a number field $K$ an imaginary quadratic field in which a prime $p$ splits into $p=\mathfrak{p}\mathfrak{p}^*$. It is conjectured in the book of Coates and Sujatha "Galois Cohomology of Elliptic Curves" (Conjecture 2.5) that the corank of the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$ extension of $K$ is $2$ when E has potentially supersingular reduction at $\mathfrak{p}$ and $\mathfrak{p}^*$, $1$ if $E$ has potentially supersingular reduction at one of the primes $\{\mathfrak{p},\mathfrak{p}^*\}$ and $0$ otherwise. The analog of this conjecture is settled for elliptic curves over $\mathbb{Q}$ Part1.
I wonder what the expected corank should be for the split prime $\mathbb{Z}_p$ extensions over an imaginary quadratic field K in which p splits into $\mathfrak{p}\mathfrak{p}^*$? What about over the anticyclotomic $\mathbb{Z}_p$ extension? (The split prime $\mathbb{Z}_p$ extension at $\mathfrak{p}$ is the $\mathbb{Z}_p$ extension of $K$ ramified only at $\mathfrak{p}$.)