  Given $\beta\in(1,2],$ the $\beta$-transformation $T_\beta: x\mapsto \beta
x\pmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle
et al.~(2019). They described the set-valued bifurcation set
  \[
  \mathcal E_\beta:=\{t\in[0, 1): K_\beta(t')\ne K_\beta(t)~\forall t'>t\},
  \]
  where $K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\ge t~\forall n\ge 0\}$ is the
survivor set. In this paper we investigate the dimension bifurcation set
  \[
  \mathcal B_\beta:=\{t\in[0, 1): \dim_H K_\beta(t')\ne \dim_H
K_\beta(t)~\forall t'>t\},
  \]
  where $\dim_H$ denotes the Hausdorff dimension.
  We show that if $\beta\in(1,2]$ is a multinacci number then the two
bifurcation sets $\mathcal B_\beta$ and $\mathcal E_\beta$ coincide. Moreover
we give a complete characterization of these two sets. As a corollary of our
main result we prove that for $\beta$ a multinacci number we have
$\dim_H(\mathcal E_\beta\cap[t, 1])=\dim_H K_\beta(t)$ for any $t\in[0, 1)$.
This confirms a conjecture of Kalle et al.~for $\beta$ a multinacci number.
