  The long-standing quest to determine the superconducting order of
Sr$_2$RuO$_4$ (SRO) has received renewed attention after recent nuclear
magnetic resonance (NMR) Knight shift experiments have cast doubt on the
possibility of spin-triplet pairing in the superconducting state. As a putative
solution, encompassing a body of experiments conducted over the years, a
$d+ig$-wave order parameter caused by an accidental near-degeneracy has been
suggested [S. A. Kivelson et al., npj Quantum Materials $\bf{5}$, 43 (2020)].
Here we develop a general Ginzburg--Landau theory for multiband
superconductors. We apply the theory to SRO and predict the relative size of
the order parameter components. The heat capacity jump expected at the onset of
the second order parameter component is found to be above the current threshold
deduced by the experimental absence of a second jump. Our results tightly
restrict theories of $d+ig$ order, and other candidates caused by a
near-degeneracy, in SRO. We discuss possible solutions to the problem.
