The counterexample conjecture for symbolic-power asymptotic regularity

Let $R={\Bbbk}[x,y,a,b]$ be the polynomial ring from the preceding construction, let $S=R[z]$, and set $I=Q\cap(f,z)\subseteq S$. The source specifies that $\operatorname{Ass}(I)=\operatorname{Min}(I)$, so the symbolic powers $I^{(n)}$ may be defined using either associated or minimal primes. Counterexample conjecture. The limit

$$\lim_{n\rightarrow\infty}\frac{\operatorname{reg} I^{(n)}}{n}$$

does not exist. This would provide a counterexample to the open question of Herzog, Hoa and Trung; the source notes that the proposed lower-bound behavior is known along the sequence $n=3\cdot2^s$, while the complementary upper-bound inequality remains open.