If there is a conflict at the workplace there is no standardized way to approach it because conflicts are not standardized. The importance of free speech is that the nuances of speech obviously reflect the nuances of experience and being. Here are some general considerations that can be seen in shades in all conflict.
If you feel there is a conflict that is present, voice it and tell the truth. Say exactly what it is you are observing, what it makes you feel, and how you think it impacts the workplace and mission. This is not the time to bring up solutions, this is a time to tell the truth. Essentially, the goal is just to verify if what you are feeling is valid or accurate to what is really happening. It may be the case that you are missing a critical element of what you are witnessing but that is inconsequential. Regardless of whether there is a conflict or not the behavior of the people around you should have momentarily shifted to address what you said. If it is misfounded or ungrounded, they will correct you and tensions should not have risen if you are only communicating what you understood to be the conflict at its highest resolution. If there are no attempts made to correct your understanding or investigate further, your coworkers and superiors are explicitly communicating that they do not care for the truth which you have communicated. If it was as simple as saying you're wrong, someone would have come forward. They are choosing not to acknowledge what is happening.
If and when there is a conflict that is being neglected unjustly and/or unreasonably you have to be willing to leave over this no matter how small before you are able to fully commit to attempting to resolve it. The resolution may not occur with your peers or at all. Sometimes this will be the only thing you can do after voicing it first. The nature of the conflict may be one sided, that is to say you may be the only one who feels it is a conflict. In this case if you cannot arbitrarily make the decision that it is resolved then you will be constantly in a state of unjust compromise. It isn’t easy or convenient, but I think having a lateral move available at all times in an infinitely useful utility. Not to blackmail your employer or coworkers with your leaving but to show you really mean what you say. And you have to mean what you say otherwise it is you who is not being truthful. To be able to walk away completely demonstrates that you know you are telling the truth and you expect the same from the people around you.
As the bureaucracy of the corporate workplace is ever expanded upon, one has to remember the principles of ethics are not defined by what is and isn’t censored. The workplace should not become a mini police-state where we all become each other’s prison guards. To voice the truth as you understand it is the first step in broadening your understanding. If your understanding is flawed, how would you seek to establish a better understanding if no one including yourself is able to articulate it?