Generating-series conjecture for the self-dual \tilde{A}_1 quiver

Let $Q=(\bullet\rightrightarrows\bullet)$ be the self-dual $\tilde{A}_1$ quiver with its two vertices exchanged by the contravariant involution, and write $\tilde{A}_1^{u,v}$ for the indicated vertex and edge signs. Let $\mathbb{L}$ be the Lefschetz motive, $q$ a formal variable, and $\mathrm{J}^{\mathrm{sd}}$, $\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}$, and $\mathrm{DT}^{\mathrm{sd}}$ the motivic, numerical, and Donaldson--Thomas self-dual invariants. Generating-series conjecture for the self-dual $\tilde{A}_1$ quiver. In the situation above, the following identities should hold:

For $\tilde{A}_1^{+,++}$ and $\tilde{A}_1^{-,--}$,

$$\sum_{n=0}^{\infty}\mathrm{J}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\frac{(1-q\mathbb{L})^{1/2}}{(1-q^{1/2})(1-q^{1/2}\mathbb{L})},$$ $$\sum_{n=0}^{\infty}\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\frac{(1+q^{1/2})^{1/2}}{(1-q^{1/2})^{3/2}},\qquad \sum_{n=0}^{\infty}\mathrm{DT}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\frac{(1-q^{1/2})^{1/2}}{(1+q^{1/2})^{3/2}}.$$

For $\tilde{A}_1^{+,+-}$ and $\tilde{A}_1^{-,+-}$, all three generating series are equal to

$$\sum_{n=0}^{\infty}\mathrm{J}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\sum_{n=0}^{\infty}\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\sum_{n=0}^{\infty}\mathrm{DT}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\frac{(1+q^{1/2})^{1/2}}{(1-q^{1/2})^{1/2}}.$$

For $\tilde{A}_1^{+,--}$ and $\tilde{A}_1^{-,++}$,

$$\sum_{n=0}^{\infty}\mathrm{J}^{\mathrm{sd}}_{(n,n)}q^{n/2}=(1-q\mathbb{L}^{-1})^{1/2},$$ $$\sum_{n=0}^{\infty}\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{(n,n)}q^{n/2}=\sum_{n=0}^{\infty}\mathrm{DT}^{\mathrm{sd}}_{(n,n)}q^{n/2}=(1-q)^{1/2}.$$

These identities are suggested by numerical computations for the two-arrow quiver; the supplied source gives no resolution status.