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We present results of the magnetic susceptibility on high-quality singlecrystals of EuPd$_2$(Si$_{1-x}$Ge$_x$)$_2$ for Ge concentrations 0 $\leq x\leq$ 0.105 performed under varying hydrostatic (He-gas) pressure 0 $\leq p\leq$ 0.5 GPa. The work extends on recent studies at ambient pressuredemonstrating the drastic change in the magnetic response fromvalence-change-crossover behavior for $x$ = 0 and 0.058, to long-rangeantiferromagnetic (afm) order below $T_{\text{N}}$ = 47 K for $x$ = 0.105. Thevalence-change-crossover temperature $T'_{\text{V}}$ shows an extraordinarilystrong pressure dependence of d$T'_{\text{V}}$/d$p$ = +(80 $\pm$ 10) K/GPa. Incontrast, a very small pressure dependence of d$T_{\text{N}}$/d$p \leq$ +(1$\pm$ 0.5) K/GPa is found for the afm order upon pressurizing the $x$ = 0.105crystal from $p$ = 0 to 0.05 GPa. Remarkably, by further increasing thepressure to 0.1 GPa, a drastic change in the ground state from afm order tovalence-change-crossover behavior is observed. Estimates of the electronicentropy, derived from analyzing susceptibility data at varying pressures,indicate that the boundary between afm order and valence-change crossoverrepresents a first-order phase transition. Our results suggest a particulartype of second-order critical endpoint of the first-order transition for $x$ =0.105 at $p_{\text{cr}} \approx$ 0.06 GPa and $T_{\text{cr}} \approx$ 45 Kwhere intriguing strong-coupling effects between fluctuating charge-, spin- andlattice degrees of freedom can be expected.

Pressure study on the interplay between magnetic order and valence-change crossover in EuPd$_2$(Si$_{1-x}$Ge$_x$)$_2$