  The affine Yangian of $\mathfrak{gl}_1$ is isomorphic to the universal
enveloping algebra of $\mathcal{W}_{1+\infty}$ and can serve as a building
block in the construction of new vertex operator algebras. In [1], a
two-parameter family generalization of $\mathcal{N}=2$ supersymmetric
$\mathcal{W}_{\infty}$ algebra was constructed by "gluing" two affine Yangians
of $\mathfrak{gl}_1$ using operators that transform as $(\square,
\overline{\square})$ and $(\overline{\square}, \square)$ w.r.t. the two affine
Yangians. In this paper we realize a similar (but non-isomorphic) two-parameter
gluing construction where the gluing operators transform as $(\square,
\square)$ and $(\overline{\square}, \overline{\square})$ w.r.t. the two affine
Yangians. The corresponding representation space consists of pairs of plane
partitions connected by a common leg whose cross-section takes the shape of
Young diagrams, offering a more transparent geometric picture than the previous
construction.
