No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient ${\mathcal{A}}_{\mathcal{D}}:=\mathcal{D}\mathrm{/}\mathsf{I}\mathsf{d}(S)$, where $\mathsf{I}\mathsf{d}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:={x \vdash y-x\dashv y \mid x,y\in $\mathcal{D}$}$,iscalledtheassociativealgebraassociatedto$$\mathcal{D}$$\mathrm{.}HereweshowthattheGelfand--Kirillovdimensionof$$\mathcal{D}$$isboundedabovebytwicetheGelfand--Kirillovdimensionof$$\mathcal{A}$_$\mathcal{D}$$. Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.