BUG: Modin's dtypes cache contains invalid value after an unsuccessful cast (when .astype(..., errors="ignore"))

[dchigarev]

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Reproducible Example

from modin.pandas.test.utils import create_test_dfs

md_df1, pd_df1 = create_test_dfs({"col1": ["a", 0, 1]})

md_res = md_df1.astype(int, errors="ignore") 
pd_res = pd_df1.astype(int, errors="ignore")

print(f"modin dtypes:\n{md_res.dtypes}\n") # int64 (successful cast?)
print(f"pandas dtypes:\n{pd_res.dtypes}\n") # object (unsuccessful as expected)

print(f"modin's insides:\n{md_res._to_pandas().dtypes}") # object (unsuccessful as expected)

Issue Description

Modin seems to change the dtypes anyway even if the cast was unsuccessful.

Expected Behavior

To work as pandas

Error Logs

Replace this line with the error backtrace (if applicable).

Installed Versions

Replace this line with the output of pd.show_versions()

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Note: this issue is a part of an epic #3804