We show that ${\mathrm{MA}}_{\mathrm{\kappa }}$ implies that each collection of $P_{\mathfrak{c}}$-points of size at most κ which has a $P_{\mathfrak{c}}$-point as an $\mathrm{RK}$ upper bound also has a $P_{\mathfrak{c}}$-point as an $\mathrm{RK}$ lower bound.