I feed round rolls...
Instead of laying it flat, we keep ours on end...
We do not remove the baler netting either but use the net to hold the shape of the roll and restrict the horses from diving in deep.
View attachment 1196490
We do not net our roll either but it is covered by a tarp to keep it out of the weather.
We direct where the horse is going to eat from with how we release the tarp and pull it back exposing the hay.
We have the horses eating from the edge in and only when about 3/4 of the roll is gone is their head "inside"...
We cut and remove netting as is needed...it is never a danger of being ingested.
For us, once we are at the point of the horses putting their face in, well, they are still withdrawing their head as they eat yet...
The pile of dropped proves just how much they grab a mouthful and leave the "inside" of the roll...
At this consumed point though honestly the roll is now headed inside the barn storage area and we feed it out by pitchfork amounts and the animals are off the ground on mats eating...that reduces the inhalation issue of head buried inside. We are able to "peel" off the roll remains and carry it out to the horses placing sheets of hay for them to eat.
We are also fussy that our hay is "HORSE" hay meaning it is cut a bit higher in the field, not at the dirt line like is done on cow rolls. That cuts down on amount of dirt that is picked up in the grasses cut.
Our haymen also double fluffs his horse hay at a minimum. If you've ever seen them fluffing and putting into rows the hay you know how much dust and dirt is in the air....it does make a difference.
Horse hay is baled lower moisture levels than cow hay, so less mold is inside or chance of it developing in dry hay versus a slightly damper baled roll for cows. Our horse hay is also barn kept, never gets wet from cutting, to dry to fluffed and baled/rolled.
My hay at center of the roll is not musty smelling, nor puff white dust nor does the bottom of the bale have these tendencies...if you are seeing any white dust, musty odor or blatant mold you need to get off of rounds as they are not being rolled for a horses consumption. It also is diligence on you to watch for, sniff carefully and be very vigilant that the hay is ground elevated and covered from moisture from the heavens or absorbed out of the ground.
Some coughing or blowing the nose is normal done to clear the dust from the sinuses. Whether square baled or round roll fed noses are blown, throats can be cleared....
Our fly masks with nose cover is proof of how often noses are blown and how much ick comes out.
My horses are currently on pasture....they blow their noses frequently as hot & dusty...they blow the nose clear of irritants.
I clear my nose & throat when dusty environments are occurring does not mean I have asthma...and I don't!
It is frequency and when that cough is done, how often, how many and does it stop or continue.
It also makes a difference in when riding does the animal continue to cough, have a discharge and difficulty resuming a "rest period" respiration rate and how quickly is the return to normal....
The horse clearing their airway of irritants..to me is a..
✔ 
Some "asthma" can be a precursor to other larger issues.
Some is just normal horse exhalation and environmental stress influenced.
Some needs far more intervention and different feeding protocols followed.
Asthma is diagnosed by:
Diagnosis of equine asthma is achieved via respiratory tract endoscopy. An endoscope is inserted up your horses nose and into your horse's windpipe. With this, the vet can see from the nostril down to where the trachea splits to go into each of the two lungs and up into the guttural pouches.
If you are dealing with a firm diagnosis of asthma from the vet, have been given instructions on how to administer medications then a different method to feeding your horse should of been discussed so you understand the risks of roll versus baled hay and if wetting baled is also suggested or needed.
You need more information from your vet with clear understanding of what's happening.
🐎... jmo.