If any applicant feels aggrieved by the action of the commission in refusing to examine him, or after examination in refusing to certify him as eligible, the commission shall, at the request of such applicant, appoint a date, time and place for a public hearing, at which time the applicant may appear, by himself or by counsel or in person and with counsel, and the commission shall then review its refusal to make such examination or certification, and testimony shall be taken. The commission shall subpoena, at the expense of the applicant, any competent witnesses requested by him. After such review, the commission shall file in its records the testimony taken and shall again make a decision, which decision shall be final and not subject to judicial review, but under no circumstances shall the provisions of this article be construed, in the case of a refusal to examine an applicant for promotion or to certify an applicant as eligible for promotion, as depriving such applicant of his right to seek a writ of mandamus, if the application for such writ is made within twenty days from the date of the decision (made after hearing as aforesaid) refusing to examine or to certify him as eligible for promotion.