fix: fix SII implementation for `NonlinearSolveNoInitCache` #310

Job	Run time
downgrade (1.10, core)	2m 15s
test (core, 1.10, ubuntu-latest)	16m 59s
downgrade (1.10, downstream)	2m 16s
test (core, 1.10, macos-latest)	13m 7s
downgrade (1.10, wrappers)	2m 16s
test (core, 1.10, windows-latest)	27m 37s
test (core, 1, ubuntu-latest)	27m 50s
test (core, 1, macos-latest)	24m 11s
test (core, 1, windows-latest)	40m 51s
test (downstream, 1.10, ubuntu-latest)	11m 32s
test (downstream, 1.10, macos-latest)	7m 26s
test (downstream, 1.10, windows-latest)	15m 53s
test (downstream, 1, ubuntu-latest)	22m 27s
test (downstream, 1, macos-latest)	12m 29s
test (downstream, 1, windows-latest)	27m 10s
test (wrappers, 1.10, ubuntu-latest)	10m 13s
test (wrappers, 1.10, macos-latest)	7m 21s
test (wrappers, 1.10, windows-latest)	13m 30s
test (wrappers, 1, ubuntu-latest)	20m 15s
test (wrappers, 1, macos-latest)	13m 40s
test (wrappers, 1, windows-latest)	22m 53s
	5h 42m 11s

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