Save the design as .PES and then copy and save as .PES 6 and see if your EM will recognize the .PES 6 format.
UPDATE -
I found this information in the Designer's Gallery Studio Plus software instruction manual under "Conversion".
"One of the most popular formats for home embroidery is the .PES file.
There are at present many versions of this file.
There are three conversion options that are unique to PES:
(1) Convert using PES version 5 with real thread colors. This allows the .PES 5 format to be written, which is necessary for the larger-field machines such as Ellageo and the EMP6
(2) Ensures trims and tie-offs. If the design already contains tie-offs, not extras are added.
(3) Convert 180mm x 300mm designs with PES 6.0 format. This is used for machines such as the Baby Lock Ellegante which require a Palette 6.0 to edit 200 x 300 designs. If you do not have this machine, turn this feature off.
If you have a file that does not require the 180mm x 300mm hoop, then the file will be written according to the PES version 5 format preference (#1, above)."
I have a Baby Lock Ellegante and just saved a design in both PES and PES 6 format (which are two of several options I have with Designer's Gallery).
I see no difference in the formats, and both opened on the EM. The design was larger than 5 x 7.
.PES 6 is a stitch format and not design software. The format has nothing to do with PE Design or PED Basic.
Brother confuses the issue by using PE this and that when naming their software products.
All software conversion and digitizing products will save to many formats, including .PES