I suppose you could consider the impact on the electrode potentials for systems 2 and 3 using the following logic.
[AuCl4]^- (aq) + 3e^- <=> Au (s) + 4Cl^- (aq) (E° = +1.00 V)
NO3^- (aq) + 4H^+ (aq) + 3e^- <=> NO (g) + 2H2O (l) (E° = +0.96V)
E°(cell) = (+0.96 V) - (+1.00 V) = -0.04 V
The electrode potential values given in the table are only correct if they are measured under standard conditions. As such, if the conditions were standard, a mixture of HCl and HNO3 would fail to dissolve gold.
However, under standard conditions, E°(cell) is only ever so slightly more negative than zero and so adjusting the conditions could make E(cell) positive and thus the reaction can be made spontaneous.
They’ve told you that the concentrations of each acid are much greater than the standard concentration of (circa) 1.00 mol dm^-3, so the conditions are most certainly non-standard. Let’s now consider each equilibrium separately:
[AuCl4]^- (aq) + 3e^- <=> Au (s) + 4Cl^- (aq) (E° = +1.00 V)
By using a large [HCl], you ensure a large [Cl^-] and so drive this equilibrium left. This means that E in practice is less -ve than the standard value given by some amount x because the equilibrium goes to the side that produces electrons. Let’s say that E = (+1.00 - x) V.
NO3^- (aq) + 4H^+ (aq) + 3e^- <=> NO (g) + 2H2O (l) (E° = +0.96V)
By using a large [HNO3], you ensure a large [NO3^-] and a large [H^+]. This drives this equilibrium right and so E increases by some amount y. Thus let’s say E = (+0.96 + y) V.
E(cell) = (+0.96 + y) V - (+1.00 - x) V = (x + y - 0.04) V, which is positive
So Au can dissolve in aqua regia in practice