Yemen's army has the Red Sea coast back for now, but Taiz is still cut off

The story
Two weeks into a big push against the Houthis, Yemen's government forces say they have won back most of the stretch of Red Sea coast they lost in September. The question now is whether they can keep it.
The Houthis caught everyone off guard last month. In about three weeks they swept down the southwest coast and took Mokha, the Dhubab area and Perim Island, all sitting right next to the Bab al Mandab strait, one of the busiest shipping lanes on the planet. Oil markets jumped.
The government answered on October 4, when presidential council chief Rashad al Alimi announced an all out campaign to retake the whole country, Sanaa included. Under heavy Saudi air cover, troops quickly reclaimed the Dhubab airfield and said they had captured Mokha. Their spokesman later claimed more than a thousand Houthi fighters were killed in a single day. Those numbers can't be checked independently.
The Houthis haven't gone away, though. They are firing missiles at the roads that keep government troops on the coast supplied, and they can still hit ships with drones from higher ground inland. Who really controls the waterway is still being argued by both sides.
The biggest worry is Taiz. While the coast fight raged, the Houthis pushed deeper into the countryside around the city and cut every supply road linking it to government held areas. For the people living there, the coastal wins haven't changed much yet.
So far the offensive looks strong on the map. Whether it lasts depends on keeping those supply lines open and on how long Saudi Arabia keeps the jets flying.