TBH i doubt it, the way blizzard designs there game, there is actually no possible way for dungeons to stay relevant unless the new dungeon gear is as good as mythic gear.

If we look back at Vanilla/tbc/wrath there was many reasons why dungeons were relevant throughout the x-pack.

For vanilla the biggest reasons why dungeons were always relevant is because it would take a LONG time to get full raid gear, not even talking about BiS just regular raid gear, you had 2-3 pieces drop per a boss and to split them between 40 people. So getting full BiS-pre-raid was extremely important back than, as a lot of that gear will stick around for months to come. Also dungeons in general back than just took a lot more time, you could probably easily do 3 dungeons at WoD launch and just barely finish 1 vanilla or TBC dungeon, they were made a lot easier in Wotlk though.

For TBC/wotlk it was because there was always new dungeons with each raid and each dungeon gave badges, that were used to buy pretty good gear. also TBC/Wrath raids only dropped 2-3 pieces as well per a boss (besides tier bosses in wrath which dropped 4-5 IIRC) so once again it would take a while to attain full raid gear, but not nearly as long as it was in vanilla. making dungeons a whole lot more valuable. The same is mostly true for cata, but this is the time when blizzard made every raid boss drop 4-5+ pieces, so you would easily get rid of your dungeon gear. but they were still relevant throughout the x-pack because of valor points and new dungeons every new patch.

That is why i highly doubt legion dungeons will stay relevant, after the initial 2 weeks(when raids start), you simply out gear dungeons and they no longer give any meaningful reward, you can easily get full bis dungeon gear within 1 week or less of grinding depending on how much you play.

Unless dungeons are going to give mythic quality gear or if they drop valor badges again that you can use to buy mythic quality gear, i highly doubt they will stay relevant. I know the new infinite challenge mode dungeons(?) will give better gear the higher tier you get, but it looks like that is only meant for higher skilled players, unless blizzard ends up making it real easy.