## Materials and Methods
### Syntheses of HESO-Samples
#### 2.2.2 Syntheses by Electrospinning  
In syntheses via electrospinning (ES), two different polymer/solvent solutions were evaluated, namely PAN/N,N-dimethylformamide (DMF) and PVP/DMF+ethanol (EtOH). In the latter case, the spinnable solution was obtained by dissolving PVP (25 wt%) in EtOH and the metal salts were separately dissolved in DMF. Both solutions were magnetically stirred at 350 rpm at room temperature (RT) until they became clear. Then, they were mixed under continuous stirring (350 rpm) for 2 h. The PAN/DMF solution was prepared by dissolving PAN (7.1 wt%) in DMF. Then, proper amounts of the metallic salts were added (Figure S1a), one at a time, under continuous stirring (350 rpm) until a homogeneous spinnable solution containing the selected equimolar combination of five metals was obtained (about 1 h). Regardless of the metal combination selected, the total amount of metals was kept constant (38.5 wt% relative to PAN and 24.7 wt% relative to PVP). After slow mixing overnight at RT, the solution was ready for deposition.  
Each homogeneous spinnable solution obtained was loaded into a 20 mL syringe equipped with a 40 mm long 0.8 mm gauge stainless-steel needle. Electrospinning (Figure S1c2) was carried out, at 22 \(\pm\) 1 \({}^{\circ}\)C and 40% relative air humidity, by using a CH-01 Electro-spinner 2.0 (Linari Engineering s.r.l.). The solution feeding rate, applied DC voltage and needle-collector distance were fixed at 23.5 \(\mu\)L min\({}^{-1}\), 15 kV and 11 cm, respectively. The pristine polymer/metal salts nanofibers (NFs) were ejected from the needle and collected over an aluminum foil to form a non-woven fibrous membrane.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multicolumn{6}{c}{**Selected Element Combination**} & \multicolumn{6}{c}{**Sample Codes**} \\ \hline \multirow{2}{*}{**Mg**} & \multirow{2}{*}{**Ti**} & \multirow{2}{*}{**Cr**} & \multirow{2}{*}{**Mn**} & \multirow{2}{*}{**Fe**} & \multirow{2}{*}{**Co**} & \multirow{2}{*}{**Ni**} & \multirow{2}{*}{**Cu**} & \multirow{2}{*}{**Zn**} & **Single/Main Oxide Phase** & & & **ES** \\ \cline{5-6} \cline{8-10}  & & & & & & & & **PVP** & **PAN** \\ \hline \multirow{5}{*}{**Mg**} & \multirow{5}{*}{**Ti**} & \multirow{5}{*}{**Cr**} & \multirow{5}{*}{**Mn**} & \multirow{5}{*}{**Fe**} & \multirow{5}{*}{**Co**} & \multirow{2}{*}{**Ni**} & \multirow{2}{*}{**Cu**} & \multirow{2}{*}{**Zn**} & **\(\mathrm{Mg_{0.2}Te_{0.2}Cu_{0.2}Zn_{0.2}})\mathrm{O_{4}}\)** & 5 MA-sg & – & – \\  & & & & & &  
After drying at RT overnight to remove the solvent residuals, the as-spun membrane was peeled from the collector and calcined in static air (Figure S1(c2)). A four-step heating process was operated in the case of PVP-derived HESO-samples (350 \({}^{\circ}\)C for 2 h, 500 \({}^{\circ}\)C for 2 h, 800 \({}^{\circ}\)C for 2 h and 900 \({}^{\circ}\)C); PAN-derived HESO-samples were obtained through a two-step heat-treatment (700 \({}^{\circ}\)C for 2 h and 900 \({}^{\circ}\)C for 2 h). The obtained powders consisted of porous architectures, generally built up by HESO-NFs (Figure S1(d2)) with spinel structure as a unique/primary phase (Figure S1e).  
In addition to pure HEO-samples, a composite carbon/HESO sample was prepared. For this purpose, the same PAN/DMF-based solution as for the corresponding pure HESO was used, but after the as-spun membrane had been detached from the collector, it was first stabilized in air at 225 \({}^{\circ}\)C [42] for 2 h, and subsequently carbonized under inert atmosphere through the same two-step heat-treatment as for pure HESO samples, so that the oxide grains formed in the composite NFs experienced the same temperatures as in those of pure oxide.