This is a spur of the moment algebraic number theory question prompted by a side remark I made in a course I'm teaching:

Let $K$ be a number field. The (Hilbert) **class field tower** of $K$ is the sequence defined by $K^0 = K$ and for all $n \geq 0$, $K^{n+1}$ is the Hilbert class field of $K^n$. Put $K^{\infty} = \bigcup_n K^n$. We say that the class field tower is infinite if $[K^{\infty}:K] = \infty$ (equivalently $K^{n+1} \supsetneq K^n$ for all $n$). Golod and Shafarevich gave examples of number fields with infinite class field tower, and thus which admit everywhere unramified extensions of infinite degree. It is now known that a number field with "sufficiently many ramified primes" has infinite class field tower.

My question is this: is there a known algorithm which, upon being given a number field, decides whether the Hilbert class field tower of $K$ is infinite?