## 2 Materials and Methods  
### Solid-State Reaction (SSR) Synthesis Protocol and Post-Synthesis Processing  
Na\({}_{3}\)Zr\({}_{2}\)Si\({}_{2}\)PO\({}_{12}\) was synthesized according to the conventional SSR protocol, obtained by a thorough comparison of procedures found in the literature [40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55]. Precursors were selected according to availability without special considerations for particle size [54; 55], in order to comply with the typical selection of starting materials employed conventionally. Thus, the SSR precursors were: Na\({}_{2}\)CO\({}_{3}\) (98%, 0.5 mm granulates, Alfa Aesar | Thermo Fisher Scientific, Waltham, MA, USA), SiO\({}_{2}\) (0.2 to 0.7 mm granulates, Umicore, Belgium), ZrO\({}_{2}\) (99%, 5 \(\upmu\)m average particle size, MilliporeSigma (Burlington, MA, USA) | Sigma-Aldrich (St. Louis, MO, USA) | Merck (Kenilworth, NJ, USA)), and NH\({}_{4}\)H\({}_{2}\)PO\({}_{4}\) (0.5 mm granulates, Scharlab, Barcelona, Spain), in the molar ratio of 1.5:2:2:1 (corresponding to the stoichiometric ratio of 3:2:2:1 in terms of Na:Si:Zr:P). No fillers or additives for sodium content enhancement or grain size refinement were added [46; 47; 48; 49; 50; 51; 52; 53; 54; 55], instead relying on ball-milling and post-synthesis processing to produce materials of appropriate quality.  
The precursors were loaded in zirconia holders and mixed with ethanol (96%, laboratory reagent grade, MilliporeSigma | Sigma-Aldrich | Merck, USA) to prepare a slurry for wet-grinding in planetary ball-mills (wet-grinding increases the specific surface area of hard agglomerated powders). Ball-milling was performed in three consecutive steps of 3 h each at 300 rpm, using zirconia balls of the recommended loading and diameters of 10, 5, and 3 mm in succession. The collected powders were dried at 80 \({}^{\circ}\)C, preheated at 600 \({}^{\circ}\)C for 4 h to remove any volatile species, and then calculated at 1100 \({}^{\circ}\)C for 4 h under dry nitrogen in a tubular furnace. This was followed by a final dry-grinding step at 500 rpm for 2 h using 3 mm zirconia balls, and a drying step at 80 \({}^{\circ}\)C as before.  
The collected fine powders were pressed into pellet form by cold isostatic pressing (RT) for 5 min using a conventional hydraulic press to form uniaxial pellets of \(\sim\)16 mm diameter and \(\sim\)1.4 mm thickness (final dimensions varied based on applied pressure levels and material loading in the stainless-steel tubular holder, with a maximum recorded size reduction of 20% for some samples discarded after quality screening). Varying pressure levels were applied to gauge the response of the material, with the initial considerations being mechanical stability and density. In order to eliminate triviality and redundancy from the dataset, results are presented for the selected pressure levels at 245 MPa (\(\sim\)2500 kg cm\({}^{-2}\)), 345 MPa (\(\sim\)3500 kg cm\({}^{-2}\)), and 540 MPa (\(\sim\)5500 kg cm\({}^{-2}\)). The manufactured pellets were returned to the tubular furnace for sintering at 1250 \({}^{\circ}\)C for 10 h, at the heating rate of 5 \({}^{\circ}\)C min\({}^{-1}\) over dry nitrogen flux. A commercially procured NASICON membrane (4 TOONE Ltd., Ulsan, Republic of Korea), of the same composition and with dimensions of 12 mm diameter and 0.9 mm thickness, was used as the reference for all subsequent analysis.