Aveira,
In a letter written while traveling, the Vilna Gaon directed various members of his family to strengthen themselves in specific areas relevant to them individually. When his mother wondered why she wasn’t mentioned, the Gaon replied that her high level of tznius made such directives unnecessary. The letter did not state that “tznius does for a woman what Torah does for a man”, nor did he make this statement in any of his Halachic writings.