  We analyze a system of three two-dimensional nonlinear Schr\"odinger
equations coupled by linear terms and with the cubic-quintic
(focusing-defocusing) nonlinearity. We consider two versions of the model:
conservative and parity-time ($\mathcal{PT}$) symmetric. These models describe
triple-core nonlinear optical waveguides, with balanced gain and losses in the
$\mathcal{PT}$-symmetric case. We obtain families of soliton solutions and
discuss their stability. The latter study is performed using a linear stability
analysis and checked with direct numerical simulations of the evolutional
system of equations. Stable solitons are found in the conservative and
$\mathcal{PT}$-symmetric cases. Interactions and collisions between the
conservative and $\mathcal{PT}$-symmetric solitons are briefly investigated, as
well.
