The star-shapedness of a generalized numerical range

Let $\mathcal{H}_n$ be the set of all $n\times n$ Hermitian matrices and $\mathcal{H}^m_n$ be the set of all $m$-tuples of $n\times n$ Hermitian matrices. For $A=(A_1,...,A_m)\in \mathcal{H}^m_n$ and for any linear map $L:\mathcal{H}^m_n\to\mathbb{R}^\ell$, we define the $L$-numerical range of $A$ by $$W_L(A):=\{L(U^*A_1U,...,U^*A_mU): U\in \mathbb{C}^{n\times n}, U^*U=I_n\}.$$ In this paper, we prove that if $\ell\leq 3$, $n\geq \ell$ and $A_1,...,A_m$ are simultaneously unitarily diagonalizable, then $W_L(A)$ is star-shaped with star center at $L\left(\frac{\mathrm{tr} A_1}{n}I_n,...,\frac{\mathrm{tr} A_m}{n}I_n\right)$.