Use Exit Access Stair for Total Travel Distance?

A small core area of a two-story office building has two stairs. One is a traditional enclosed exit stair fire-rated at 1-hour. The other stair is intended to be an "open access stair". The project is under NFPA 101 for means of egress and is sprinkler protected under NFPA 13.

NFPA 101 Chapter 38 New Business Occupancies
   NFPA 101 38.3 Protection
      NFPA 101 38.3.1 Protection of Vertical Openings
         NFPA 101 38.3.1.1 Vertical openings shall be enclosed or protected in accordance with Section 8.6, unless otherwise permitted by any of the following: (4) Exit access stairs in accordance with 38.2.4.6 shall permitted to be unenclosed.

         NFPA 101 38.2.4.6 A single means of egress shall be permitted for a maximum two-story, single-tenant space or building provided that both of the following criteria are met:
            (1) The building is protected throughout by an approved, supervised automatic sprinkler system in accordance with 9.7.1.1(1).
            (2) The total travel to the outside does not exceed 100 ft.

The only way to have an unenclosed stair for this building is if the "total travel to the outside" can include travel through the unenclosed "exit access stair". 

Is it permissible to have one open stair and one enclosed stair with "total travel to the outside" going through the open access stair?

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