The Shulchon Aruch Orach Chaim Siman 268 S'eef 2,when discussing the Tefilos on Shabbos, says "אם טעה והתחיל תפלת חול גומר אותה ברכה",meaning that if someone started saying one of the weekday Brochos in Shemona Esrai he should already finish it. Why? Since in truth we should said the weekday Berochos on Shabbos (on Shabbos we also need Daas "אם אין דעת הבדלה הבדלה מנין").
The Chachomim,however,in order not to strain us too much on Shabbos chose to skip them .However if you started one of them already then finish it and with Kovono.
Now let us turn to the issue at hand,but not without first citing the' Levush'in OC siman 133 who lets us in on the secret of what transpires upon the opening of the Aron."שהוא לנו במקום ביאת מקדש לפני ולפנים" WOW! WE ARE IN THE KODSHAI HAKODOSHIM!
Now, returning to the point,if someone were to start saying Ashrei when the Aron is still open then it would be correct to finish the entire Ashrei with the Aron open, as Ashrei in the Bais Hamikdash לפני ולפנים is 'What An Opportunity',and so much more powerful and ideal,and the only reason we keep the Aron closed the entire Davening is simply because we are not ready to behave the entire time in the manner that would be due in such a lofty place.But if you started then finish it in the ideal place it should be said.Therefor people had a Minhag not to start Ashrai in the first place cto avoid running into the issue of needing to finish it with the Aron open.
In conclusion we don't want to start another section with the Aron open, so as not to dilute the feeling of the Kodesh Kodoshim(that one can say anything without paying attention)and so that one would not have to finish the next section with it open.
let us from now on realize where we are standing when the Aron is open and act accordingly.