When someone says "They highly suspect this will work..", do they think it will work or do they think it will not work?
They highly suspect this will work.
While this is open to some interpretation, it would normally be taken to mean that they strongly believe it will work.
From Merriam-Webster's definition of the verb suspect:
3 : to imagine to exist or be true, likely, or probable • I suspect he's right.
: to imagine something to be true or likely
In the following examples sentences, suspect can be replaced with believe:
I suspect [believe] it will rain.
Call the doctor immediately if you suspect [believe] you've been infected.
The latest research confirms what scientists have long suspected [believed].
I suspect [believe] she's not who she says she is.
There are other senses of suspect that have different meanings and which are used in different contexts.
Merriam-Webster also provides this sense of suspect:
1 : to imagine (one) to be guilty or culpable on slight evidence or without proof • suspect him of giving false information
This is very similar to believe, but in a specific legal context.
The police do not suspect [believe it to be] murder in this case.
The fire chief suspects [believes it to be] arson.
Finally, Merriam-Webster provides one more sense of suspect:
2 : to have doubts of : DISTRUST • suspects her motives
In other words, it's used to express doubt as to the quality of someone or something.
This "doubt" sense of suspect is commonly based on syntactical context.
I suspect her sincerity.
This means I doubt her sincerity.
It's more often meant when it's used with a pronoun and an adjective or noun: suspect his beliefs, suspect her credibility, suspect its advisability, suspect their motives.
Other syntactical constructions more often have the "believe" sense of suspect.
In the question's example sentence, suspect is not followed by a pronoun and adjective or noun, so it's more likely to have the "believe" sense.
You have provided a little context, but still it can be answered.
There are couple of other meanings of suspect, but in given context following will work fine.
To suspect is to doubt or to question (something). In your case, it looks more like a doubt, then question.
When you doubt something, you are uncertain (of something) or lack conviction (in something). When you question something, you doubt about the truth or validity (of something).
In your assertion,
"They highly suspect this will work.." (sic)
It shows lack of conviction, hence doubt, that it will work. But, to say, that it will work or not is entirely depend upon the context and who is saying it.
Cambridge dictionary defines it as:
suspect (verb) (DOUBT)
to not trust; to doubt:
I suspect her honesty/loyalty.
We suspected his motives in making his offer.