Invariants Heuristics for anti-cyclotomic extensions

Let $p$ be an odd prime. For an imaginary quadratic field $K$ in which $p$ does not split, let $K_{\mathrm{ac}}/K$ be the anti-cyclotomic $\mathbb{Z}_p$-extension and let $\lambda$ and $\mu$ denote its Iwasawa invariants.

Invariants Heuristics. Among such imaginary quadratic fields, one always has

$$\mu=0,$$

and the proportion of fields for which $\lambda=0$ is at least

$$\left(1-\frac{1}{p}\right)+\frac{1}{p-1}\prod_{j\geq 1}\left(1-p^{-j}\right).$$

The claim is proposed from computational evidence and is motivated by the preceding intersection heuristic and known results in the cyclic class-group case; the source does not prove the asserted universal vanishing or the stated lower bound.